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Your data matches 135 different statistics following compositions of up to 3 maps.
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Matching statistic: St001903
St001903: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 1
[1,2] => 2
[2,1] => 0
[1,1,1] => 1
[1,1,2] => 1
[1,2,1] => 2
[2,1,1] => 0
[1,1,3] => 2
[1,3,1] => 1
[3,1,1] => 0
[1,2,2] => 2
[2,1,2] => 0
[2,2,1] => 1
[1,2,3] => 3
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 1
[1,1,1,1] => 1
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,2,1,1] => 2
[2,1,1,1] => 0
[1,1,1,3] => 1
[1,1,3,1] => 2
[1,3,1,1] => 1
[3,1,1,1] => 0
[1,1,1,4] => 2
[1,1,4,1] => 1
[1,4,1,1] => 1
[4,1,1,1] => 0
[1,1,2,2] => 1
[1,2,1,2] => 2
[1,2,2,1] => 2
[2,1,1,2] => 0
[2,1,2,1] => 0
[2,2,1,1] => 1
[1,1,2,3] => 1
[1,1,3,2] => 2
[1,2,1,3] => 2
[1,2,3,1] => 3
[1,3,1,2] => 1
[1,3,2,1] => 1
[2,1,1,3] => 0
[2,1,3,1] => 1
[2,3,1,1] => 0
[3,1,1,2] => 0
[3,1,2,1] => 0
Description
The number of fixed points of a parking function.
If $(a_1,\dots,a_n)$ is a parking function, a fixed point is an index $i$ such that $a_i = i$.
It can be shown [1] that the generating function for parking functions with respect to this statistic is
$$
\frac{1}{(n+1)^2} \left((q+n)^{n+1} - (q-1)^{n+1}\right).
$$
Matching statistic: St001498
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 85%●distinct values known / distinct values provided: 80%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 85%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [.,.]
=> [1,0]
=> ? = 1
[1,1] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> ? ∊ {0,2}
[1,2] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> ? ∊ {0,2}
[2,1] => [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[1,1,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,1,2,3}
[1,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,1,2,3}
[1,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 0
[2,1,1] => [2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,1,2,3}
[1,3,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 0
[3,1,1] => [2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,2,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,1,2,3}
[2,1,2] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[2,2,1] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,1,2,3}
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 0
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2
[3,1,2] => [2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1
[1,1,1,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,4}
[1,1,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,4}
[1,1,2,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,2,1,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 0
[2,1,1,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,4}
[1,1,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,3,1,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 0
[3,1,1,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,4}
[1,1,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,4,1,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 0
[4,1,1,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,2,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,4}
[1,2,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,2,2,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,1,1,2] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,2,1] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,2,1,1] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,4}
[1,1,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,3,1,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,3,2,1] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,1,1,3] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,1] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,3,1,1] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,1,1,2] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,2,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,2,1,1] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,4}
[1,1,4,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,2,4,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,4,1,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,4,2,1] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,1,1,4] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,1] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,4,1,1] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,1,1,2] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,1,2,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,2,1,1] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,3,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,4}
[1,3,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,3,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,4}
[1,2,2,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,4}
[1,2,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,4}
[1,2,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,4}
[1,2,3,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,4}
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,3,4}
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St001124
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 79%●distinct values known / distinct values provided: 60%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 79%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1,0]
=> []
=> ? = 1
[1,1] => [1,2] => [1,0,1,0]
=> [1]
=> ? ∊ {0,1,2}
[1,2] => [1,2] => [1,0,1,0]
=> [1]
=> ? ∊ {0,1,2}
[2,1] => [2,1] => [1,1,0,0]
=> []
=> ? ∊ {0,1,2}
[1,1,1] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,1,2] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1
[2,1,1] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 0
[1,1,3] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 0
[3,1,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> ? ∊ {1,2,2,2,3}
[1,2,2] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1
[2,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 0
[2,2,1] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? ∊ {1,2,2,2,3}
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 0
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 0
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? ∊ {1,2,2,2,3}
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> ? ∊ {1,2,2,2,3}
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> ? ∊ {1,2,2,2,3}
[1,1,1,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,1,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,1,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,2,1,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[2,1,1,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[1,1,1,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,1,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,3,1,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[3,1,1,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,1,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,1,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[1,4,1,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[4,1,1,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,1,2,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,2,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,2,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[2,1,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[2,1,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[2,2,1,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[1,1,2,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,1,3,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,2,1,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,2,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,3,1,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[1,3,2,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[2,1,1,3] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[2,1,3,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[2,3,1,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[3,1,1,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[3,1,2,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[3,2,1,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[1,1,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,1,4,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[1,2,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,2,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[1,4,1,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[1,4,2,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[2,1,1,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[2,1,4,1] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[3,3,1,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,4,1,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[4,3,1,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,2,2,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,2,3,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,3,2,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,2,2,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,2,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,4,2,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[4,2,2,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,3,3,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,2,3,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,3,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,3,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St001657
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 77%●distinct values known / distinct values provided: 60%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 77%●distinct values known / distinct values provided: 60%
Values
[1] => [1,0]
=> [1,0]
=> []
=> ? = 1
[1,1] => [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 0
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {1,2}
[2,1] => [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {1,2}
[1,1,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 0
[1,1,2] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,2,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[2,1,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[1,3,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[3,1,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[1,2,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[2,1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[2,2,1] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,2,2,2,3}
[1,3,2] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,2,2,2,3}
[2,1,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,2,2,2,3}
[2,3,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,2,2,2,3}
[3,1,2] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,2,2,2,3}
[3,2,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,2,2,2,3}
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 0
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 0
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 0
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 0
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 0
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[3,2,1,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,1,2,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,1,4,2] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,2,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,2,4,1] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,4,1,2] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,4,2,1] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[2,1,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[2,1,4,1] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[1,2,4,3] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[1,3,2,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[1,3,4,2] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[1,4,2,3] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[1,4,3,2] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,1,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,1,4,3] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,3,1,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,3,4,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,4,1,3] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,4,3,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,1,2,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,1,4,2] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,2,1,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,2,4,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,4,1,2] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,4,2,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[4,1,2,3] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[4,1,3,2] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[4,2,1,3] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[4,2,3,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[4,3,1,2] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[4,3,2,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
Description
The number of twos in an integer partition.
The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].
Matching statistic: St001632
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 60% ●values known / values provided: 70%●distinct values known / distinct values provided: 60%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 60% ●values known / values provided: 70%●distinct values known / distinct values provided: 60%
Values
[1] => [1,0]
=> [1,0]
=> ([],1)
=> ? = 1
[1,1] => [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> ([],2)
=> ? ∊ {0,2}
[2,1] => [1,0,1,0]
=> [1,1,0,0]
=> ([],2)
=> ? ∊ {0,2}
[1,1,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,1,3}
[1,1,2] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 1
[1,3,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 1
[3,1,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 1
[1,2,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 2
[2,1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 2
[2,2,1] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> ? ∊ {0,0,0,0,0,1,3}
[1,3,2] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> ? ∊ {0,0,0,0,0,1,3}
[2,1,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> ? ∊ {0,0,0,0,0,1,3}
[2,3,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> ? ∊ {0,0,0,0,0,1,3}
[3,1,2] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> ? ∊ {0,0,0,0,0,1,3}
[3,2,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> ? ∊ {0,0,0,0,0,1,3}
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 0
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 0
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 0
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 0
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,2,1,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,2,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[1,1,4,2] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[1,2,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[1,2,4,1] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[1,4,1,2] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[1,4,2,1] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[2,1,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[2,1,4,1] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[2,4,1,1] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[4,1,1,2] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[4,1,2,1] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[4,2,1,1] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[1,1,3,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,1,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,2,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,2,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,2,1,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,2,2,1] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,4,3] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,2,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,4,2] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,2,3] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,3,2] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,4,3] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,3,1,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,3,4,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,4,1,3] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,4,3,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,1,2,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,1,4,2] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,2,1,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,2,4,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,4,1,2] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,4,2,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,1,2,3] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,1,3,2] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,2,1,3] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,2,3,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,3,1,2] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,3,2,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St000512
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 66%●distinct values known / distinct values provided: 60%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 66%●distinct values known / distinct values provided: 60%
Values
[1] => [1,0]
=> [1,0]
=> []
=> ? = 1
[1,1] => [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,1,2}
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,1,2}
[2,1] => [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,1,2}
[1,1,1] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 0
[1,1,2] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[1,2,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[2,1,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[1,1,3] => [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,3,1] => [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[3,1,1] => [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,2,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,1,1,1,1,2,2,2,3}
[2,1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,1,1,1,1,2,2,2,3}
[2,2,1] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,1,1,1,1,2,2,2,3}
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3}
[1,3,2] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3}
[2,1,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3}
[2,3,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3}
[3,1,2] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3}
[3,2,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2,3}
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[3,2,1,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,1,2,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,1,4,2] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,2,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,2,4,1] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,4,1,2] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,4,2,1] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[2,1,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[2,1,4,1] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[2,4,1,1] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[4,1,1,2] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[4,1,2,1] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[4,2,1,1] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,2,3,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[1,3,2,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[1,3,3,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,1,3,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,3,1,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,3,3,1] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,1,2,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,1,3,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,2,1,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,2,3,1] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,3,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,3,2,1] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[1,2,4,3] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[1,3,2,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[1,3,4,2] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[1,4,2,3] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[1,4,3,2] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,1,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,1,4,3] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,3,1,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,3,4,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,4,1,3] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[2,4,3,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,1,2,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,1,4,2] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,2,1,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,2,4,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,4,1,2] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[3,4,2,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[4,1,2,3] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[4,1,3,2] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[4,2,1,3] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[4,2,3,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[4,3,1,2] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
[4,3,2,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4}
Description
The number of invariant subsets of size 3 when acting with a permutation of given cycle type.
Matching statistic: St000929
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 66%●distinct values known / distinct values provided: 40%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 66%●distinct values known / distinct values provided: 40%
Values
[1] => [[]]
=> [1,0]
=> []
=> ? = 1
[1,1] => [[],[]]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,1,2}
[1,2] => [[[]]]
=> [1,1,0,0]
=> []
=> ? ∊ {0,1,2}
[2,1] => [[[]]]
=> [1,1,0,0]
=> []
=> ? ∊ {0,1,2}
[1,1,1] => [[],[],[]]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0
[1,1,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[1,2,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3}
[2,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[1,1,3] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[1,3,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,0,1,2,2,2,3}
[3,1,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[1,2,2] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[2,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3}
[2,2,1] => [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[1,2,3] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,0,1,2,2,2,3}
[1,3,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3}
[2,1,3] => [[[[]]]]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3}
[2,3,1] => [[[[]]]]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3}
[3,1,2] => [[[[]]]]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3}
[3,2,1] => [[[],[]]]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,0,1,2,2,2,3}
[1,1,1,1] => [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
[1,1,1,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[1,1,2,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,2,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[2,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[1,1,1,3] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[1,1,3,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[3,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[1,1,1,4] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[1,1,4,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,4,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[4,1,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[1,1,2,2] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[1,2,1,2] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[1,2,2,1] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[2,1,1,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[2,1,2,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[2,2,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[1,1,2,3] => [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[1,1,3,2] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,2,1,3] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,3,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[1,3,1,2] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,2,1] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,1,3] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[2,1,3,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,3,1,1] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[3,1,1,2] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[3,1,2,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,2,1,1] => [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[1,1,2,4] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,1,4,2] => [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[1,2,1,4] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,4,1] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,1,2] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,2,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[2,1,1,4] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[2,1,4,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,4,1,1] => [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[4,1,1,2] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[4,1,2,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,2,1,1] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[1,1,3,3] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[1,3,1,3] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[1,3,3,1] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[3,1,1,3] => [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[3,1,3,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[3,3,1,1] => [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[1,1,3,4] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,1,4,3] => [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,3,1,4] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,4,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[1,4,1,3] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,3,1] => [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[2,1,2,3] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,3,2,1] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,4,2] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,2,4] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,4,2] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,4,2,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,2,1,2] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,2,3] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,3,1,3] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,1,2,3] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,1,3,2] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,2,3,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,4,3] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,4,2] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,3,2] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,3,4] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,3,1,4] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,4,3,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,1,2,4] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,2,4,1] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,4,2,1] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,1,2,3] => [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,2,1,3] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,3,1,2] => [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St000137
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000137: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 59%●distinct values known / distinct values provided: 40%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000137: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 59%●distinct values known / distinct values provided: 40%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 1
[1,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,2}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,2}
[2,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1,2}
[1,1,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[1,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[1,2,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[2,1,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,1,3] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[3,1,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[2,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[2,2,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[1,3,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[2,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[2,3,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[3,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[3,2,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[1,1,1,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,2,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,1,1,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,1,3] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,3,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,1,1,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,1,4] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,4,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,4,1,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,1,1,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,2,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,2,2,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,1,2,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,2,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,1,2,3] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,3,2] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,2,3,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,3,2,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,1,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,1,3,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,3,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,1,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[3,1,2,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[3,2,1,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,4] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,4,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,2,1,4] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,4,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,4,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,2,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,4,1,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,1,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[4,1,2,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[4,2,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,1,3,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,3,1,3] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,3,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,1,1,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[3,1,3,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[3,3,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,1,3,4] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,1,4,3] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,1,4] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,4,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,1,3] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,3,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,1,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[3,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[3,4,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[4,1,1,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[4,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[4,3,1,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,2,2,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,2,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,2,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,2,2,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,2,3] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,3,2] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,2,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,1,3,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,2,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,2,3,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,3,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,3,2,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,2,4] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,4,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,2,4,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,4,2,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,3,3] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,3,2] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
Description
The Grundy value of an integer partition.
Consider the two-player game on an integer partition.
In each move, a player removes either a box, or a 2x2-configuration of boxes such that the resulting diagram is still a partition.
The first player that cannot move lose. This happens exactly when the empty partition is reached.
The grundy value of an integer partition is defined as the grundy value of this two-player game as defined in [1].
This game was described to me during Norcom 2013, by Urban Larsson, and it seems to be quite difficult to give a good description of the partitions with Grundy value 0.
Matching statistic: St001122
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001122: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 59%●distinct values known / distinct values provided: 40%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001122: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 59%●distinct values known / distinct values provided: 40%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 1
[1,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,2}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,2}
[2,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1,2}
[1,1,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[1,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[1,2,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[2,1,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,1,3] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[3,1,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[2,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[2,2,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[1,3,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[2,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[2,3,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[3,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[3,2,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[1,1,1,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,2,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,1,1,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,1,3] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,3,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,1,1,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,1,4] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,4,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,4,1,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[4,1,1,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,2,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,2,2,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,1,2,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,2,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,1,2,3] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,3,2] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,2,3,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,3,2,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,1,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,1,3,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,3,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,1,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[3,1,2,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[3,2,1,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,4] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,4,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,2,1,4] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,4,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,4,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,2,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,4,1,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,1,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[4,1,2,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[4,2,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,1,3,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,3,1,3] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,3,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[3,1,1,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[3,1,3,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[3,3,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,1,3,4] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,1,4,3] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,1,4] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,4,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,1,3] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,3,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,1,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[3,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[3,4,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[4,1,1,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[4,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[4,3,1,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,2,2,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,2,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,2,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,2,2,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,2,3] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,3,2] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,2,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,1,3,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,2,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,2,3,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,3,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,3,2,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,2,4] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,4,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,2,4,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[2,4,2,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,3,3] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,3,2] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4}
Description
The multiplicity of the sign representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{1^n}$, for $\lambda\vdash n$. It equals $1$ if and only if $\lambda$ is self-conjugate.
Matching statistic: St001247
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001247: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 60%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001247: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 1
[1,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,2}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,2}
[2,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1,2}
[1,1,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[1,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[1,2,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[2,1,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,1,3] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[3,1,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[2,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[2,2,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[1,3,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[2,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[2,3,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[3,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[3,2,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2,2,2,3}
[1,1,1,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,2,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,1,1,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,1,3] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,3,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,1,1,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,1,4] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,4,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,4,1,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[4,1,1,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,2,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,2,2,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[2,1,2,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,2,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,1,2,3] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,3,2] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,2,3,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,3,2,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,1,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[2,1,3,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,3,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,1,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[3,1,2,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[3,2,1,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,4] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,1,4,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,2,1,4] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,4,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,4,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,2,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[2,4,1,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,1,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[4,1,2,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[4,2,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,1,3,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,3,1,3] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,3,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[3,1,1,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[3,1,3,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[3,3,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,1,3,4] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,1,4,3] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,1,4] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,4,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,1,3] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,4,3,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,1,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[3,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[3,4,1,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[4,1,1,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[4,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[4,3,1,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,2,2,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[2,1,2,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[2,2,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,2,2,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,2,3] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,3,2] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,2,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[2,1,3,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,2,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,2,3,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[2,3,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,3,2,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,2,4] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,4,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[2,2,4,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[2,4,2,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,2,3,3] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
[1,3,3,2] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,4}
Description
The number of parts of a partition that are not congruent 2 modulo 3.
The following 125 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001249Sum of the odd parts of a partition. St001383The BG-rank of an integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001568The smallest positive integer that does not appear twice in the partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001389The number of partitions of the same length below the given integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001527The cyclic permutation representation number of an integer partition. St001541The Gini index of an integer partition. St001571The Cartan determinant of the integer partition. St001587Half of the largest even part of an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001933The largest multiplicity of a part in an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000618The number of self-evacuating tableaux of given shape. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000781The number of proper colouring schemes of a Ferrers diagram. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001128The exponens consonantiae of a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000993The multiplicity of the largest part of an integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000456The monochromatic index of a connected graph. St000706The product of the factorials of the multiplicities of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001570The minimal number of edges to add to make a graph Hamiltonian. St000454The largest eigenvalue of a graph if it is integral. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000260The radius of a connected graph. St001651The Frankl number of a lattice.
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